What is the Tsiolkovsky rocket equation?

Delta-v = Isp × g0 × ln(m0 / mf). It gives the total change in velocity a rocket can produce from its exhaust velocity (Isp × g0) and its mass ratio (initial mass over final mass). It ignores gravity and drag losses, so real ascents need more delta-v than the equation returns.

Why does delta-v depend on the logarithm of the mass ratio?

Each kilogram of propellant must accelerate all the propellant still on board. Integrating thrust over a continuously shrinking mass gives a natural logarithm, so doubling the mass ratio adds a fixed delta-v rather than doubling it. That is why staging works: dropping empty tanks resets the ratio.

How much propellant do I need for a given delta-v?

Propellant mass = mf × (exp(delta-v / ve) − 1), where mf is the dry mass including payload and ve = Isp × g0. The inverse calculator on this page solves it directly and also reports the propellant mass fraction.

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Tsiolkovsky Rocket Equation

Delta-v from specific impulse and wet/dry mass, or the propellant needed to reach a target delta-v. Reports exhaust velocity, mass ratio, propellant mass fraction, liftoff thrust-to-weight, and burn time. Ideal (no gravity or drag losses); see the notes below for how much to add for a real ascent.

Fly it: this equation is the engine under Apoapsis, a free in-browser 3D rocket and orbital-mechanics sim by the same author. The default numbers below are the sim's Reliant-class stage. Build one, launch, and compare the Δv readout to this page.

Δv from a stage

Specific impulse, I_sps

Initial (wet) mass, m₀kg (stage + propellant + payload)

Final (dry) mass, m_fkg (everything left at burnout)

Thrust, FkN (optional; for TWR and burn time)

Local gravity, gm/s² (for TWR only; 1.62 on the Moon)

Effective exhaust velocity, v_e—m/s

Mass ratio, m₀/m_f——

Delta-v, Δv—m/s (ideal)

Propellant mass—kg

Propellant mass fraction, ζ—m_p / m₀

Liftoff thrust-to-weight—F / (m₀ g); must exceed 1 to lift off

Mass flow rate, ṁ—kg/s

Burn time—s at full throttle

Defaults: Isp 310 s, 30,000 kg wet, 8,000 kg dry, 215 kN thrust. Set thrust to 0 to skip TWR and burn time.

Tsiolkovsky (ideal) rocket equation:

$$ \Delta v = v_e \ln\frac{m_0}{m_f} = I_{sp}\, g_0 \ln\frac{m_0}{m_f} $$

Propellant mass fraction, thrust-to-weight, and burn time:

$$ \zeta = \frac{m_0 - m_f}{m_0} \qquad \mathrm{TWR} = \frac{F}{m_0\, g} \qquad t_b = \frac{m_0 - m_f}{\dot m} = \frac{(m_0 - m_f)\, v_e}{F} $$

Δv ideal velocity change · v_e effective exhaust velocity · I_sp specific impulse · g₀ standard gravity, 9.80665 m/s² (a defined constant, used even off Earth) · m₀ initial mass · m_f final mass · ṁ propellant mass flow · F thrust · g local gravity.

Propellant for a target Δv

Target Δvm/s

Specific impulse, I_sps

Dry mass incl. payload, m_fkg

Required mass ratio—m₀/m_f = e^Δv/v_e

Propellant required—kg

Wet mass, m₀—kg

Propellant mass fraction——

Defaults: a 3,200 m/s upper-stage burn at Isp 345 s carrying 2,500 kg dry. Fractions above about 0.90 are hard to build; that limit is why rockets stage.

Inverting the rocket equation:

$$ \frac{m_0}{m_f} = e^{\Delta v / v_e} \qquad m_p = m_f\left(e^{\Delta v / v_e} - 1\right) $$

What the equation does and does not include

The rocket equation is exact for a vehicle in free space with constant exhaust velocity. It says nothing about the trajectory. A real ascent spends Δv fighting gravity while the vehicle climbs, pushing through the atmosphere, and steering, so the budget to reach a given orbit is always larger than the orbital speed itself. For Earth, low orbit is roughly 7,800 m/s of orbital velocity, but launch vehicles budget on the order of 9,300 to 9,500 m/s once gravity, drag, and steering losses are added. Treat this page as the ideal number, then add losses appropriate to the ascent profile.

Why staging works

Δv scales with the logarithm of the mass ratio, so pushing a single stage past a mass ratio of about 10 (propellant fraction 0.90) buys less and less. Dropping an empty tank and engine part-way up resets the ratio for the next stage. Two stages with mass ratio 4 each deliver the same Δv as one stage with mass ratio 16, and mass ratio 16 is not something you can build out of tanks that have to survive launch loads.

Specific impulse reference values

Approximate published values, sea level / vacuum, in seconds. Use the manufacturer's figure for design work.

Propellant / engine class	I_sp sea level	I_sp vacuum
Solid motor (APCP)	~240	~270
Kerosene / LOX gas-generator (Merlin 1D class)	~280	~310
Methane / LOX full-flow staged combustion (Raptor class)	~330	~350
Hydrogen / LOX staged combustion (RS-25 class)	~365	~450
Hydrogen / LOX expander upper stage (RL10 class)	—	~450
Hydrazine monopropellant thruster	—	~220
Ion thruster (xenon)	—	~3,000

Sea-level values are lower because ambient pressure pushes back on the nozzle exit. Vacuum values apply above roughly 30 km and in orbit.

Units and g₀

Specific impulse in seconds is exhaust velocity divided by standard gravity, g₀ = 9.80665 m/s². That constant is a unit conversion, not the local gravity, so it stays the same on the Moon or in deep space. The thrust-to-weight output on this page uses the separate local-gravity input, which is where the body you are standing on matters.

References: Tsiolkovsky, K. E. (1903). Exploration of Outer Space by Means of Rocket Devices. Sutton, G. P. and Biblarz, O. (2017). Rocket Propulsion Elements, 9th ed., Wiley, ch. 2 and 4. Curtis, H. D. (2020). Orbital Mechanics for Engineering Students, 4th ed., Butterworth-Heinemann, ch. 11.

Related resources

Apoapsis — free 3D rocket and orbital flight sim (Board Gaming Hub)
Ideal gas law calculator (tank pressurization)
Eclipse sun position
Linear interpolator (Isp tables)

Tsiolkovsky Rocket Equation

Δv from a stage

Propellant for a target Δv

What the equation does and does not include

Why staging works

Specific impulse reference values

Units and g0

Related resources

Monthly engineering case studies

Units and g₀